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yesterday I asked a question and we get the answer, for reference this is what we ask

Problem 1 : Let $(X,M,\mu)$ be a measure space and $f$ is a real-valued function on $X$ such that $$\int_X |f| d\mu <\infty$$ . Then for any $\epsilon >0$ we can find a measurable set $E$ such that $\mu(E) <\infty$ and $$ \int_{X \backslash E} |f| d\mu <\epsilon.$$

There is another question which asked before,for reference this is the question :-

Probelm 2 : Find $\delta >0$ such that $\int_E |f| d\mu < \epsilon$ whenever $\mu(E)<\delta$

My question: Is there is a way can go from the first problem to the second one? which we should directly use the first problem in proving the second one.

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henry
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  • Problem 1 assumes $\int_X |f|<\infty$. That's not assumed in the linked question. – David Mitra Dec 29 '14 at 20:41
  • If you assume $f$ is integrable, consider the functions $f_n=\max{|f|, n}$. $(f_n)$ is a non-decreasing sequence with pointwise limit $|f|$. So $\int f_n$ converges to $\int |f|$. Note $\int_E |f|=\int_E (|f|-f_n)+\int_E f_n$ and that for a fixed $n$, $\int_E f_n$ is small if $\mu(E)$ is small enough. – David Mitra Dec 29 '14 at 20:51
  • So, no, the two problems are very different. – David Mitra Dec 29 '14 at 20:54

1 Answers1

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As mentioned by David Mitra, the two problems are rather different. The first property states that $f$ is small enough, so that it takes small values on sets of infinite measure - a rough analogy on $\Bbb R^n$ would be that $f(x)\to 0$ as $x\to\infty$. In contrast, the second property says that the measure $\mathrm d\nu = f\;\mathrm d\mu$ is absolutely continuous w.r.t. $\mu$ which of course uses the fact that $f$ is not incredibly big but in essence the issues here may arise even if $\int f\mathrm d\mu$ is finite. Yet again, roughly it means that the first property is violated if $f = \infty$ on positive $\mu$ sets, whereas the second is violated if $f = \infty$ on null $\mu$ sets.

That said, in math one can rarely really safely claim that two results are completely unrelated.

SBF
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  • thanks for your answer,I know in math most of problem are can be related in some way, but in this problem I meant directly or heavily,but your answer give more insight about the two problem. – henry Dec 30 '14 at 15:27